Inverse Geometric Problem Research Articles

Evolutionary-numerical algorithm for unsteady inverse geometric problems in double-connected domains

Numerical solution of the problem of reconstruction of the inner boundary of the double-connected domain from the given Cauchy data on the outer part of the domain, for the heat and wave equations is considered. The inverse problem is reformulated as a minimization of the nonlinear functional. A real-valued genetic algorithm is used for the minimization. A tness function of the individual is proposed, for the calculation of which it is necessary to solve the non-stationary Dirichlet problem. For this problem, rst a semi discretization by the time variable is performed using the Rothe's method, and then the method of fundamental solutions is applied to the obtained recurrent sequence of stationary inhomogeneous problems. The proposed approach is easy to extend to the case of higher dimensions, therefore two dimensional and three dimensional domains are considered. The algorithm is tested on several examples for both equations and the stability of the method is con rmed for the noised input data.

Method for prediction magnetic silencing of uncertain energy-saturated extended technical objects in prolate spheroidal coordinate system

Aim. Development of method for prediction by energy-saturated extended technical objects magnetic silencing based on magnetostatics geometric inverse problems solution and magnetic field spatial spheroidal harmonics calculated in prolate spheroidal coordinate system taking into account of technical objects magnetic characteristics uncertainties. Methodology. Spatial prolate spheroidal harmonics of extended technical objects magnetic field model calculated as magnetostatics geometric inverse problems solution in the form of nonlinear minimax optimization problem based on near field measurements for prediction far extended technical objects magnetic field magnitude. Nonlinear objective function calculated as the weighted sum of squared residuals between the measured and predicted magnetic field COMSOL Multiphysics software package used. Nonlinear minimax optimization problems solutions calculated based on particle swarm nonlinear optimization algorithms. Results. Results of prediction extended technical objects far magnetic field magnitude based on extended technical objects spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system using near field measurements with consideration of extended technical objects magnetic characteristics uncertainty. Originality. The method for prediction by extended technical objects magnetic cleanliness based on spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of magnetic characteristics uncertainty is developed. Practical value. The important practical problem of prediction extended technical objects magnetic silencing based on the spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of extended technical objects magnetic characteristics uncertainty solved. References 48, figures 2.

Reconstruction of the shape of a flaw in ferromagnetic plate by solving inverse problem of magnetostatics and series of direct problems

The article presents a verification technique for solving the inverse geometric problem of magnetostatics in a soft magnetic ferromagnet plate. The technique involves solving a number of direct problems, in which the shape of the defect obtained by solving the inverse geometric problem of magnetostatics is used as a first approximation, and then increasing or decreasing the depth of the defect without changing the shape of the boundary surface — comparing the topographies of the magnetic field components obtained during measurements above the plate surface and calculated (as a result of solving the direct problem) at the same points of the components of the magnetic stray field from the reconstructed three-dimensional defect. As a result of applying the technique, the geometric parameters of the defect under study can also be refined. Obtaining the initial conditions for solving the inverse problem and solving direct problems of magnetostatics is carried out using the finite element method in the ELMER program. The technique works with one-sided access to any surface of the plate (a defect-free surface or a surface with a defect).

Local geometric properties of conductive transmission eigenfunctions and applications

Abstract The purpose of the paper is twofold. First, we show that partial-data transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in $\mathbb{R}^n$ , $n=2,3$ . Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [26] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [26] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [26] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.

On some geometrical eigenvalue inverse problems involving the p-Laplacian operator

On some geometrical eigenvalue inverse problems involving the p-Laplacian operator

Solution of Inverse Geometric Problems Using a Non-Iterative MFS

Solution of Inverse Geometric Problems Using a Non-Iterative MFS

Scope of applicability of the technique for constructing magnetic induction lines for flaw defectometry of extended objects

A technique for approximate solution of the inverse geometric problem of magnetostatics for a plate made of a soft magnetic ferromagnet in a magnetic field is presented. The technique is presented both for the case of the location of magnetic transducers directly above the surface defect of loss of continuity of the metal, and for the case in which the magnetic transducers are located above the defect-free surface of the plate. It is assumed that the plate is accessible from one side only. The sizes of defects in which the proposed technique works reliably are determined. It is shown that the proposed technique can be used in mobile devices to carry out flaw detection of drill pipes using the MFL (Magnetic flux leakage) method directly at drilling sites.

Application of the topological sensitivity method to the detection of breast cancer

Abstract This paper is concerned with an approach based on the topological sensitivity notion to solve a geometric inverse problem for a linear wave equation. The considered inverse problem is motivated by elastography. More precisely, the modelli ng of our application system has been aimed toward the detection of a breast tumour, in particular, and to enable the calculation of the tumour size, location and type. We start our analysis by rephrasing the considered inverse problem as an optimization one minimizing an energy cost functional. We establish an estimation describing the asymptotic behaviour of the wave equation solution with respect to the presence of a small tumour in the breast, which plays an important role in the derivation of a topological asymptotic formula for the considered cost function. Based on the derived theoretical results, we have developed a numerical algorithm for solving our inverse problem, which requires only one iteration. Some numerical experiments are presented to point out the efficiency and accuracy of the proposed approach.

Determination of rigid inclusions immersed in an isotropic elastic body from boundary measurement

We study the determination of some rigid inclusions immersed in an isotropic elastic medium from overdetermined boundary data. We propose an accurate approach based on the topological sensitivity technique and the reciprocity gap concept. We derive a higher-order asymptotic formula, connecting the known boundary data and the unknown inclusion parameters. The obtained formula is interesting and useful tool for developing accurate and robust numerical algorithms in geometric inverse problems.

Replacing voids and localized parameter changes with fictitious forcing terms in boundary-value problems

We present a general observation on how to replace changes of material properties in limited regions within a domain with fictitious forcing terms in initial- and boundary-value problems associated with wave propagation and diffusion. Then, by considering a paradigmatic heat conduction problem on a domain with a cavity, we prove that the presence of the void can be replaced by a fictitious heat source with support contained within the cavity. We illustrate this fact in a situation where the source term can be analytically recovered from the values of the temperature and heat flux at the boundary of the cavity. Our result provides a strategy to map the nonlinear geometric inverse problem of void identification into a more manageable one, that involves the identification of forcing terms given the knowledge of external boundary data. To set the stage for a systematic study of the inverse problem, we present algebraic reconstructions, based on a finite-element discretization of the domain, that give an approximation of the fictitious source from different sets of temperature measurements. We show how the accuracy of the reconstruction is reflected on the void identification.

Stable reconstruction of simple Riemannian manifolds from unknown interior sources

Consider the geometric inverse problem: there is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a Riemannian manifold with boundary? With a finite set of sources we can only hope to get an approximate reconstruction, and we indeed provide a discrete metric approximation to the manifold with explicit data-driven error bounds when the manifold is simple. This is the geometrization of a seismological inverse problem where we measure the arrival times on the surface of waves from an unknown number of unknown interior microseismic events at unknown times. The closeness of two metric spaces with a marked boundary is measured by a labeled Gromov–Hausdorff distance. If measurements are done for infinite time and spatially dense sources, our construction produces the true Riemannian manifold and the finite-time approximations converge to it in the metric sense

A combination of Kohn-Vogelius and DDM methods for a geometrical inverse problem

We consider the inverse geometrical problem of identifying the discontinuity curve of an electrical conductivity from boundary measurements. This standard inverse problem is used as a model to introduce and study a combined inversion algorithm coupling a gradient descent on the Kohn-Vogelius cost functional with a domain decomposition method that includes the unknown curve in the domain partitioning. We prove the local convergence of the method in a simplified case and numerically show its efficiency for some two dimensional experiments.

Shape reconstructions by using plasmon resonances with enhanced sensitivity

This paper investigates the shape reconstructions of sub-wavelength objects from near-field measurements in transverse electromagnetic scattering. This geometric inverse problem is notoriously ill-posed and challenging. We develop a novel reconstruction scheme of using plasmon resonances with significantly enhanced sensitivity and resolution. First, by spectral analysis we establish a sharp quantitative relationship between the sensitivity of the reconstruction and the plasmon resonance. It shows that the sensitivity functional blows up when plasmon resonance occurs. Hence, the signal to noise ratio is significantly improved and the robustness and effectiveness of the reconstruction are ensured. Second, a variational regularization method is proposed to overcome the ill-posedness, and an alternating iteration method is introduced to automatically select the regularization parameters. Third, we use Laplace approximation method to capture the statistical information of the target scattering object. Both rigorous theoretical analysis and extensive numerical experiments are conducted to validate the promising features of our method.

Optimal shape design for a time-dependent Brinkman flow using asymptotic analysis techniques

In this paper, we consider the geometric inverse problem of recovering an obstacle ω immersed in a bounded fluid flow Ω governed by the time-dependent Brinkman model. We reformulate the inverse problem into an optimization problem using a least squares functional. We prove the existence of an optimal solution to the optimization problem. Then, we perform the asymptotic expansion of the cost function in a simple way using a penalty method. An important advantage of this method is that it avoids the truncation method used in the literature. To reconstruct the obstacle, we propose a fast algorithm based on the topological derivative. Finally, we present some numerical experiments in two- and three-dimensional cases showing the efficiency of the proposed method.

Numerical Solving of Radiation Geometrical Inverse Problem

Compared to large satellites, on-board electronics of microsatellites, such as microprocessors, are more exposed to radiation and high-energy particles. This problem can lead to single-event upsets, which might generate false commands such as thruster firing or switching heaters on or off. In some cases, these upsets cause the loss of a spacecraft orientation and endanger the whole mission. This makes important to elaborate a reliable and robust attitude determination system which will enable us to correct and recover the spacecraft’s attitude after a failure. It must have a simple design, low mass, and power consumption. In this paper we suggest a new method of determination of spacecraft’s attitude based on heat fluxes analysis that could form the basis of such a system. In order to develop it we have to solve two inverse problems sequentially. The first inverse problem is the estimating of heat fluxes absorbed by the spacecraft surface. The second one is to determine the spacecraft’s attitude based on the estimated values of radiative heat fluxes. This paper is dedicated to the solution of the second inverse problem.

Modelling, analysis, and numerical methods for a geometric inverse source problem in variable-order time-fractional subdiffusion

There exist research works on studying time-dependent integer-order and time-fractional constant-order geometric inverse source problems in the literature. The time-fractional variable-order geometric inverse source problems although also have important physical applications have not been studied mathematically and numerically in literature. The aim of this work is to study an inverse source problem associated with a variable-order time-fractional subdiffusion equation. We first build a mathematical model and show existence of the optimal shape for shape reconstruction of the source support. Then, shape sensitivity analysis is performed to propose a shape gradient optimization algorithm allowing deformations for numerically solving the model problem. In order to reconstruct the source support with topology unknown a priori, moreover, we build a phase-field model and propose a gradient algorithm allowing both shape and topological changes by a phase-field method. A variety of numerical examples are presented to demonstrate effectiveness of the two algorithms.

IDENTIFICATION OF NODAL POINTS OF ELASTIC INCLUSION IN ELASTIC PLANE

A geometric inverse problem of identifying an isotropic, linearly elastic inclusion in an isotropic, linearly elastic plane is considered. It is assumed that constant stresses are given at infinity, and the displacements and acting loads are known on some closed curve enclosing the inclusion. In the case when the inclusion is a quadrature domain, a method for identifying its nodal points has been developed. A numerical example is considered.

Research on geometric inverse problem based on the firefly conjugate gradient method

In this paper the 2-D steady-state heat transfer geometric inverse problem is solved using the finite element method, conjugate gradient method, and firefly algorithm. Based on the finite element method for solving the forward heat transfer model, and based on continuous iterative optimisation of the conjugate gradient method, the accuracy of the error function of the measured and estimated values is kept within a certain range, so that the geometry of the object under test can be calculated in an inverse way. In the study of the forward problem, the temperature field distribution is solved using circular pipes as the research objects, and the feasibility of applying the finite element method to heat transfer problems is verified. The inverse problem takes the circular tube as the object and considers two different corrosion defects on the inner wall of the circular tube. Simultaneously, the firefly algorithm is introduced based on the conjugate gradient method to stochastically optimize the temperature data and suppress the fluctuation of the inversion result. Numerical experiment results indicate that the method in this paper can perform more accurate geometric shape recognition when there is a certain temperature measurement error or the temperature measurement information is relatively complete.

An efficient moving pseudo-boundary MFS for void detection

We consider the method of fundamental solutions (MFS) for the determination of the boundary of a void. In the proposed formulation the location of the pseudo-boundary is not fixed. The MFS discretization of the corresponding inverse geometric boundary value problem yields a system of nonlinear equations in which the coefficients in the MFS approximation, the discrete radii in the polar parametrization, the coordinates of the centre of the void and the expansion and dilation coefficients of the pseudo-boundaries are unknown. For the minimization of the resulting functional we employ the nonlinear least squares minimization routine lsqnonlin from the MATLAB® optimization toolbox . In contrast to previous studies, we exploit the option which enables the user to provide the analytical expression for the Jacobian of the system, and show that, although tedious, this leads to spectacular savings in computational time. The case of multiple voids is also addressed.

Method for Solving Inverse Geometric Problem of Magnetostatics for Surface Defects of a Soft Magnetic Ferromagnet

Method for Solving Inverse Geometric Problem of Magnetostatics for Surface Defects of a Soft Magnetic Ferromagnet